NeuroPMD: Neural Fields for Density Estimation on Product Manifolds

TL;DR

NeuroPMD introduces a deep neural field for density estimation on product manifolds by parameterizing the log-density with a network that is fed by a random, data-adaptive Laplace–Beltrami eigenfunction encoding. The model globally regularizes via an LBO-based roughness penalty and employs a scalable, unbiased stochastic gradient algorithm to train on large, high-dimensional product domains. Theoretical results show deep encodings exponentially enrich the representation space on toroidal domains, while practical encoding choices (including non-separable rotations) and sinusoidal activations enable efficient training. Empirical evaluations on synthetic torus data and a real brain-connectivity dataset demonstrate superior performance over KDE, tensor-product basis methods, and beta-network variants, particularly in high-dimensional settings and anisotropic densities. The approach offers a scalable, flexible, and interpretable framework for density estimation on complex manifolds with broad applicability in neuroscience and beyond.

Abstract

We propose a novel deep neural network methodology for density estimation on product Riemannian manifold domains. In our approach, the network directly parameterizes the unknown density function and is trained using a penalized maximum likelihood framework, with a penalty term formed using manifold differential operators. The network architecture and estimation algorithm are carefully designed to handle the challenges of high-dimensional product manifold domains, effectively mitigating the curse of dimensionality that limits traditional kernel and basis expansion estimators, as well as overcoming the convergence issues encountered by non-specialized neural network methods. Extensive simulations and a real-world application to brain structural connectivity data highlight the clear advantages of our method over the competing alternatives.

Figures (13)

• Figure 1: Product manifold point process data from a neuroscience connectomics application. A) Cortical surface mesh of the left hemisphere with representative white matter fiber connections. The red zoomed-in region highlights fibers terminating on the surface. B) Observed endpoints on the surface. Each red point corresponds with a single blue point, which together form the surface coordinates of a connection. C) Observed endpoints represented under spherical parameterization.
• Figure 2: True (log) density function on $\mathbb{T}^2$ (left column) and estimates of (log) density from each method from a randomly selected experimental replication. Both rows present the same function visualized with different colorbars for enhanced comparison.
• Figure 3: Marginal density function estimates for connections from the medial orbitofrontal cortex (MOFC), generated using our method (left) and tensor product basis (right). Black dots represent the endpoints connected to the MOFC. Color scales are normalized within each image to emphasize differences in the shape of the functions.
• Figure 4: Same as Figure \ref{['fig:rda_marg_density']}, with marginal density functions mapped to the cortical surface. Top Row: The MOFC is highlighted in red in both medial (left) and lateral (right) views. Middle Row: Comparing the estimated marginal density functions by the NeuroPMD and TPB method. Bottom Row: Close-up views of the marginal density functions mapped to the cortical surface. Endpoints of fiber curves connecting to the MOFC are marked in black.
• Figure S1: Two non-separable eigenfunctions (left column) and separable eigenfunctions (right column) of $\Delta_{\mathbb{T}^2}$ for the same eigenspace $\lambda = 20$.

Theorems & Definitions (22)

• Remark 1
• Theorem 1
• Proposition 1
• Proposition 2
• proof
• proof
• Lemma S1
• proof : Proof of Lemma \ref{['LEM:pD']}
• Remark 2
• Lemma S2